I found a reference to Mary Barnes in a publication that can be accessed through ancestry.com, "A catalogue of the names of the first Puritan settlers of the colony of Connecticut". It was printed in 1846 as a collection of names and events collected from state and town records. "Mary Barnes, of Pequett" was listed on p. 15 (online page counter).
Most exciting was a listing for John Andrews Sr, p. 111. He was from Hartford, d. 1681, w. Mary. Sons: Benjamin, John, Abraham, Daniel, and Joseph. Daughters: MARY BARNES, Hannah Richards, and Rachel Buck.
Mary's husband, Thomas Barnes, was listed on pages 17 and 114
I think this provides a good clue as to her maiden name and parents. Here is the link:
http://interactive.ancestry.com/16187/dvm_LocHist004143-0006...